Solution:

There are no factors that are not paired. Hence, 256 is a perfect square. The square of an even number is always even. Hence, 256 is the square of an even number.

(ii) 324 = 2 x 2 x 3 x 3 x 3 x 3 = (2 x 2) x (3 x 3) x (3 x 3) There are no factors that are not paired. Hence, 324 is a perfect square. The square of an even number is always even. Hence, 324 is the square of an even number.

(iv) 5476 = 2 x 2 x 37 x 37 = (2 x 2) x (37 x 37) There are no factors that are not paired. Hence, 5476 is a perfect square. The square of an even number is always even. Hence, 5476 is the square of an even number.

(v) 373758 = 2 x 3 x 7 x 11 x 809 Here, each factor appears only once, so grouping them into pairs of equal factors is not possible. It means that 373758 is not the square of an even number.

Hence, the numbers that are the squares of even numbers are 256, 324, 1296 and 5476.

Question: 12

By just examining the units digit, can you tell which of the following cannot be whole squares?

(i) 1026

(ii) 1028

(iii) 1024

(iv) 1022

(v) 1023

(vi) 1027

Solution:

If the unit’s digit of a number is 2, 3, 7 or 8, the number cannot be a whole square.

(i) 1026

1026 has 6 as the unit’s digit, so it is possibly a perfect square.

(ii) 1028

1028 has 8 as the unit’s digit, so it cannot be a perfect square.

(iii) 1024

1024 has 4 as the unit’s digit, so it is possibly a perfect square.

(iv) 1022

1022 has 2 as the unit’s digit, so it cannot be a perfect square.

(v) 1023

1023 has 3 as the unit’s digit, so it cannot be a perfect square.

(vi) 1027

1027 has 7 as the unit digit, so it cannot be a perfect square.

Hence, by examining the unit’s digits, we can be certain that 1028, 1022, 1023 and 1027 cannot be whole squares.

Question: 13

Write five numbers which you cannot decide whether they are squares.

Solution:

A number whose unit digit is 2, 3, 7 or 8 cannot be a perfect square.

On the other hand, a number whose unit digit is 1, 4, 5, 6, 9 or 0 might be a perfect square (although we will have to verify whether it is a perfect square or not).

Applying the above two conditions, we cannot quickly decide whether the following numbers are squares of any numbers:

1111, 1444, 1555, 1666, 1999

Question: 14

Write five numbers which you cannot decide whether they are square just be looking at the units digit.

Solution:

A number whose unit digit is 2, 3, 7 or 8 cannot be a perfect square.

On the other hand, a number whose unit digit is 1, 4, 5, 6, 9 or 0 might be a perfect square although we have to verify that.

Applying these two conditions, we cannot determine whether the following numbers are squares just by looking at their unit digits:

1111, 1001, 1555, 1666 and 1999

Question: 15

Write True (T) and false (F) for the following statements.

(i) The number of digits in a square number is even.

(ii) The square of a prime number is prime.

(iii) The sun of two square numbers is a square number.

(iv) The difference of two square numbers is a square number.

(v) The product of two square numbers is a square number.

(vi) No square number is negative.

(vii) There is no square number between 50 and 60

(viii) There are fourteen square number up to 200.

Solution:

(i) The number of digits in a square number is even.

False

Example: 100 is the square of a number but its number of digits is three, which is not an even number.

(ii) The square of a prime number is prime.

False

If p is a prime number, its square is p2, which has at least three factors: 1, p and p2. Since it has more than two factors, it is not a prime number.

(iii) The sun of two square numbers is a square number.

False

1 is the square of a number (1 = 12). But 1 + 1 = 2, which is not the square of any number.

(iv) The difference of two square numbers is a square number.

False

4 and 1 are squares (4 = 22, 1 = 12). But 4 – 1 = 3, which is not the square of any number.

(v) The product of two square numbers is a square number.

True

If a2 and b2 are two squares, their product is a2 x b2 = (a x b)2, which is a square.

(vi) No square number is negative.

True

The square of a negative number will be positive because negative times negative is positive.

(vii) There is no square number between 50 and 60

True 72 = 49 and 82 = 64. 7 and 8 are consecutive numbers and hence there are no square numbers between 50 and 60.

(viii) There are fourteen square number up to 200.

True 142 is equal to 196, which is below 200. There are 14 square numbers below 200.